Problem: All of the 5th grade teachers and students from Gardner Bullis went on a field trip to an art museum. Tickets were $$5.00$ each for teachers and $$3.50$ each for students, and the group paid $$51.50$ in total. The next month, the same group visited a science museum where the tickets cost $$20.00$ each for teachers and $$12.50$ each for students, and the group paid $$192.50$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5x+3.5y = 51.5}$ ${20x+12.5y = 192.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-20x-14y = -206}$ ${20x+12.5y = 192.5}$ Add the top and bottom equations together. $ -1.5y = -13.5 $ $ y = \dfrac{-13.5}{-1.5}$ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into $ {5x+3.5y = 51.5}$ to find $x$ ${5x + 3.5}{(9)}{= 51.5}$ $5x+31.5 = 51.5$ $5x = 20$ $x = \dfrac{20}{5}$ ${x = 4}$ You can also plug ${y = 9}$ into $ {20x+12.5y = 192.5}$ and get the same answer for $x$ ${20x + 12.5}{(9)}{= 192.5}$ ${x = 4}$ There were $4$ teachers and $9$ students on the field trips.